Certain Subclasses of 𝑝-Valent Meromorphic Functions Associated with a New Operator

Abstract

We introduce two classes of 𝑝-valent meromorphic functions associated with a new operator and derive several interesting results for these classes.

1. Introduction

Let Σ_𝑝 denote the class of functions of the form 𝑓(𝑧)=𝑧^−𝑝+^∞_𝑛=1𝑎_𝑛−𝑝𝑧^𝑛−𝑝𝑝∈ℕ={1,2,3,…}, (1) which are analytic and 𝑝-valent in the punctured unit disc 𝑈^∗={𝑧∈ℂ∶0<|𝑧|<1}=𝑈⧵{0}. Let 𝑃_𝑘(𝛽) be the class of functions 𝜙(𝑧) analytic in 𝑈 satisfying the properties ℎ(0)=1 and₀^2𝜋ℜ{ℎ(𝑧)}−𝛽1−𝛽𝑑𝜃≤𝑘𝜋, (2) where 𝑧=𝑟𝑒^𝑖𝜃, 𝑘≥2, and 0≤𝛽<1. The class 𝑃_𝑘(𝛽)was introduced by Padmanabhan and Parvatham [1]. For 𝛽=0, the class 𝑃_𝑘(0)=𝑃_𝑘 was introduced by Pinchuk [2]. Also we note that 𝑃₂(𝛽)=𝑃(𝛽), where 𝑃(𝛽) is the class of functions with positive real part greater than 𝛽 and 𝑃₂(0)=𝑃, where 𝑃 is the class of functions with positive real part. From (2), we have ℎ(𝑧)∈𝑃_𝑘(𝛽) if and only if there exists ℎ₁,ℎ₂∈𝑃(𝛽) such that ℎ(𝑧)=𝑘4+12ℎ₁(𝑧)−𝑘4−12ℎ₂(𝑧)(𝑧∈𝑈). (3) It is known that the class 𝑃_𝑘(𝛽) is a convex set (see [3]).

For functions 𝑓(𝑧)∈Σ_𝑝 given by (1) and 𝑔(𝑧)∈Σ_𝑝 given by 𝑔(𝑧)=𝑧^−𝑝+^∞_𝑛=1𝑎_𝑛−𝑝𝑧^𝑛−𝑝𝑝∈ℕ, (4) the Hadamard product (or convolution) of 𝑓(𝑧) and 𝑔(𝑧) is defined by 𝑓∗𝑔(𝑧)=𝑧^−𝑝+^∞_𝑛=1𝑎_𝑛−𝑝𝑏_𝑛−𝑝𝑧^𝑛−𝑝=𝑔∗𝑓(𝑧). (5)

Aqlan et al. [4] defined the operator 𝑄^𝛼_𝛽,𝑝∶Σ_𝑝→Σ_𝑝 by 𝑄^𝛼_𝛽,𝑝𝑓(𝑧)=𝑧^−𝑝+Γ𝛼+𝛽Γ𝛽^∞_𝑛∑₌₁Γ𝑛+𝛽Γ𝑎𝑛+𝛽+𝛼^𝑛−𝑝𝑧^𝑛−𝑝𝛼>0;𝛽>−1;𝑝∈ℕ;𝑓∈Σ_𝑝𝑓(𝑧)𝛼=0;𝛽>−1;𝑝∈ℕ;𝑓∈Σ_𝑝. (6)

Mostafa [5] used Aqlan et al. operator and defined the following linear operator 𝐻^𝛼_{𝑝,𝛽,𝜇}∶Σ_𝑝→Σ_𝑝 as follows.

First put 𝐺^𝛼_𝛽,𝑝(𝑧)=𝑧^−𝑝+Γ𝛼+𝛽Γ𝛽^∞_𝑛=1Γ𝑛+𝛽Γ𝑧𝑛+𝛽+𝛼^𝑛−𝑝𝑝∈ℕ (7) and let 𝐺^𝛼∗_{𝛽,𝑝,𝜇} be defined by 𝐺^𝛼_𝛽,𝑝(𝑧)∗𝐺^𝛼∗_{𝛽,𝑝,𝜇}(𝑧)=1𝑧^𝑝(1−𝑧)^𝜇𝜇>0;𝑝∈ℕ. (8) Then 𝐻^𝛼_{𝑝,𝛽,𝜇}𝑓(𝑧)=𝐺^𝛼∗_𝛽,𝑝(𝑧)∗𝑓(𝑧)𝑓∈Σ_𝑝. (9) Using (7) and (9), we have 𝐻^𝛼_{𝑝,𝛽,𝜇}𝑓(𝑧)=𝑧^−𝑝+Γ𝛽Γ𝛼+𝛽^∞_𝑛=1Γ𝑛+𝛽+𝛼𝜇_𝑛Γ𝑛+𝛽(1)_𝑛𝑎_𝑛−𝑝𝑧^𝑛−𝑝, (10) where (𝜈)_𝑛 denotes the Pochhammer symbol given by (𝜈)_𝑛=Γ(𝜈+𝑛)Γ(𝜈)=1()𝜈()⋯().𝑛=0𝜈+1𝜈+𝑛−1)(𝑛∈ℕ (11) It is readily verified from (10) that (see [5]) 𝑧𝐻^𝛼_{𝑝,𝛽,𝜇}𝑓(𝑧)^′=𝛼+𝛽𝐻^𝛼+1_{𝑝,𝛽,𝜇}𝑓(𝑧)−𝛼+𝛽+𝑝𝐻^𝛼_{𝑝,𝛽,𝜇}𝑓(𝑧),𝑧𝐻^𝛼_{𝑝,𝛽,𝜇}𝑓(𝑧)^′=𝜇𝐻^𝛼_{𝑝,𝛽,𝜇+1}𝑓(𝑧)−𝜇+𝑝𝐻^𝛼_{𝑝,𝛽,𝜇}𝑓(𝑧). (12)(13)

It is noticed that by putting 𝜇=1 in (10), we obtain the operator 𝐻^𝛼_𝑝,𝛽,1𝑓(𝑧)=𝐻^𝛼_𝑝,𝛽𝑓(𝑧)=𝑧^−𝑝+Γ𝛽Γ𝛼+𝛽^∞_𝑛=1Γ𝑛+𝛼+𝛽Γ𝑎𝑛+𝛽^𝑛−𝑝𝑧^𝑛−𝑝. (14)

Now, by using the linear operator 𝐻^𝛼_{𝑝,𝛽,𝜇}𝑓(𝑧), we introduce classes of 𝑝-valent Bazilevic functions of Σ_𝑝 as follows.

Definition 1. A function 𝑓(𝑧)∈Σ_𝑝 is said to be in the class 𝐾_𝑝(𝛼,𝛽,𝜇,𝛿,𝜌,𝛾) if it satisfies the following condition: 1−𝛾𝑧^𝑝𝐻^𝛼_{𝑝,𝛽,𝜇}𝑓(𝑧)^𝛿𝐻+𝛾^{𝛼+1𝑝,𝛽,𝜇}𝑓(𝑧)𝑓(𝑧)𝐻^𝛼_{𝑝,𝛽,𝜇}𝑓(𝑧)𝑧^𝑝𝐻^𝛼_{𝑝,𝛽,𝜇}𝑓(𝑧)^𝛿∈𝒫_𝑘𝜌.𝛼,𝜇>0;𝛽>−1;𝑘≥2;𝛾≥0,𝛿>0;0≤𝜌<1;𝑝∈ℕ;𝑧∈𝑈 (15)

Definition 2. A function 𝑓(𝑧)∈Σ_𝑝 is said to be in the class 𝑀_𝑝(𝛼,𝛽,𝜇,𝛿,𝜌,𝛾) if it satisfies the following condition: 1−𝛾𝑧^𝑝𝐻^𝛼_{𝑝,𝛽,𝜇}𝑓(𝑧)^𝛿𝐻+𝛾^{𝛼𝑝,𝛽,𝜇+1}𝑓(𝑧)𝑓(𝑧)𝐻^𝛼_{𝑝,𝛽,𝜇}𝑓(𝑧)𝑧^𝑝𝐻^𝛼_{𝑝,𝛽,𝜇}𝑓(𝑧)^𝛿∈𝒫_𝑘𝜌.𝛼,𝜇>0;𝛽>−1;𝑘≥2;𝛾≥0,𝛿>0;0≤𝜌<1;𝑝∈ℕ;𝑧∈𝑈 (16)

In this paper, we investigate several properties of the classes 𝐾_𝑝(𝛼,𝛽,𝜇,𝛿,𝜌,𝛾) and 𝑀_𝑝(𝛼,𝛽,𝜇,𝛿,𝜌,𝛾).

2. Main Results

Unless otherwise mentioned, we assume throughout this paper that 𝑘≥2, 𝛼,𝛿,𝛾,𝜇>0, 𝛽>−1, 𝛼+𝛽>0, 0≤𝜌<1, 𝑝∈ℕ and all powers are understood as principle values.

To prove our results we need the following lemma.

Lemma 3 (see [6]). Let 𝑢=𝑢₁+𝑖𝑢₂, 𝑣=𝑣₁+𝑖𝑣₂ and Φ(𝑢,𝑣) be a complex-valued function satisfying the conditions:(i)Φ(𝑢,𝑣) is continuous in a domain 𝐷∈ℂ².(ii)(0,1)∈𝐷 and Φ(1,0)>0.(iii)ℜ{Φ(𝑖𝑢₂,𝑣₁)}>0 whenever (𝑖𝑢₂,𝑣₁)∈𝐷 and 𝑣₁≤−(1/2)(1+𝑢²₂).
If ℎ(𝑧)=1+𝑐_𝑛𝑧^𝑛+𝑐_𝑛+1𝑧^𝑛+1+⋯ is analytic in 𝑈 such that (ℎ(𝑧),𝑧ℎ^′(𝑧))∈𝐷 and ℜ{Φ(ℎ(𝑧),𝑧ℎ^′(𝑧))}>0 for 𝑧∈𝑈, then ℜ{ℎ(𝑧)}>0 in 𝑈.

Employing the technique used by Noor and Muhammad [7] and Aouf and Seoudy [8] for multivalent functions, we prove the following theorems.

Theorem 4. If 𝑓(𝑧)∈𝐾_𝑝(𝛼,𝛽,𝜇,𝛿,𝜌,𝛾), then 𝑧^𝑝𝐻^𝛼_{𝑝,𝛽,𝜇}𝑓(𝑧)^𝛿∈𝒫_𝑘𝜂, (17) where 𝜂 is given by 𝜂=2𝛿𝜌𝛼+𝛽+𝛾.2𝛿𝛼+𝛽+𝛾 (18)

Proof. Setting 𝑧^𝑝𝐻^𝛼_{𝑝,𝛽,𝜇}𝑓(𝑧)^𝛿(𝑧)=ℎ(𝑧)=𝑘=𝐻1−𝜂+𝜂4+121−𝜂ℎ₁(𝑧)−𝑘+𝜂4−121−𝜂ℎ₂(𝑧)+𝜂, (19) where ℎ_𝑖(𝑧)(𝑖=1,2) are analytic in 𝑈 with ℎ_𝑖(0)=1(𝑖=1,2), and ℎ(𝑧) is given by (3). Differentiating both sides of (19) with respect to 𝑧 and using (12) in the resulting equation, we obtain 1−𝛾𝑧^𝑝𝐻^𝛼_{𝑝,𝛽,𝜇}𝑓(𝑧)^𝛿𝐻+𝛾^{𝛼+1𝑝,𝛽,𝜇}𝑓(𝑧)𝑓(𝑧)𝐻^𝛼_{𝑝,𝛽,𝜇}𝑓(𝑧)𝑧^𝑝𝐻^𝛼_{𝑝,𝛽,𝜇}𝑓(𝑧)^𝛿=ℎ(𝑧)𝛾1−𝜂+𝜂+1−𝜂𝑧ℎ^′(𝑧)𝛿𝛼+𝛽∈𝑃_𝑘𝜌(𝑧∈𝑈), (20) which implies that 1ℎ1−𝜌𝜂−𝜌+1−𝜂_𝑖(𝑧)+𝛾1−𝜂𝑧ℎ^′_𝑖(𝑧)𝛿().𝛼+𝛽∈𝑃𝑧∈𝑈;𝑖=1,2 (21) We form the functional Φ(𝑢,𝑣) by choosing 𝑢=ℎ_𝑖(𝑧), 𝑣=𝑧ℎ^′_𝑖(𝑧), that is, Φ()𝛾𝑣𝑢,𝑣=𝜂−𝜌+1−𝜂𝑢+1−𝜂𝛿.𝛼+𝛽 (22) Clearly, the first two conditions of Lemma 3 are satisfied. Now, we verify the condition (iii) as follows: ℜΦ𝑖𝑢₂,𝑣₁𝛾𝑣=𝜂−𝜌+ℜ1−𝜂₁𝛿𝛾𝛼+𝛽≤𝜂−𝜌−1−𝜂1+𝑢²₂=2𝛿𝛼+𝛽𝐴+𝐵𝑢²₂,2𝐶 (23) where ,,.𝐴=2𝛿𝛼+𝛽𝜂−𝜌−𝛾1−𝜂𝐵=−𝛾1−𝜂𝐶=2𝛿𝛼+𝛽 (24) We note that ℜ{Φ(𝑖𝑢₂,𝑣₁)}<0 if and only if 𝐴=0, 𝐵<0. From 𝜂 given by (18), we have 0≤𝜂<1, 𝐴=0, and 𝐵<0. Therefore, applying Lemma 3, we have ℎ_𝑖(𝑧)∈𝑃(𝑖=1,2) and consequently 𝐻(𝑧)∈𝑃_𝑘(𝜂)for 𝑧∈𝑈. This completes the proof of Theorem 4.

Similarly, we can prove the following theorem for the class 𝑀_𝑝(𝛼,𝛽,𝜇,𝛿,𝜌,𝛾).

Theorem 5. If 𝑓(𝑧)∈𝑀_𝑝(𝛼,𝛽,𝜇,𝛿,𝜌,𝛾), then 𝑧^𝑝𝐻^𝛼_{𝑝,𝛽,𝜇}𝑓(𝑧)^𝛿∈𝒫_𝑘𝜒, (25) where 𝜒 is given by 𝜒=2𝛿𝜇𝜌+𝛾.2𝛿𝜇+𝛾 (26)

Theorem 6. If 𝑓(𝑧)∈𝐾_𝑝(𝛼,𝛽,𝜇,𝛿,𝜌,𝛾), then 𝑧^𝑝𝐻^𝛼_{𝑝,𝛽,𝜇}𝑓(𝑧)^𝛿/2∈𝒫_𝑘(𝜁), (27) where 𝜁 is given by 𝜁=𝛾+𝛾²+4𝜌𝛿𝛼+𝛽𝛿𝛼+𝛽+𝛾2𝛿.𝛼+𝛽+𝛾 (28)

Proof. Let 𝑓(𝑧)∈𝐾_𝑝(𝛼,𝛽,𝜇,𝛿,𝜌,𝛾) and 𝑧^𝑝𝐻^𝛼_{𝑝,𝛽,𝜇}𝑓(𝑧)^𝛿=𝑀(𝑧)=(1−𝜁)ℎ(𝑧)+𝜁²=𝑘4+12(1−𝜁)ℎ₁(𝑧)+𝜁²−𝑘4−12(1−𝜁)ℎ₂(𝑧)+𝜁², (29) where ℎ_𝑖(𝑧)(𝑖=1,2) are analytic in 𝑈 with ℎ_𝑖(0)=1(𝑖=1,2) and ℎ(𝑧) is given by (3). Differentiating both sides of (29) with respect to 𝑧 and using (12) in the resulting equation, we obtain 1−𝛾𝑧^𝑝𝐻^𝛼_{𝑝,𝛽,𝜇}𝑓(𝑧)^𝛿𝐻+𝛾^{𝛼+1𝑝,𝛽,𝜇}𝑓(𝑧)𝑓(𝑧)𝐻^𝛼_{𝑝,𝛽,𝜇}𝑓(𝑧)𝑧^𝑝𝐻^𝛼_{𝑝,𝛽,𝜇}𝑓(𝑧)^𝛿=(1−𝜁)ℎ(𝑧)+𝜁²+)ℎ(𝑧)×()(1−𝜁+𝜁2𝛾1−𝜁𝑧ℎ^′(𝑧)𝛿𝛼+𝛽∈𝑃_𝑘𝜌(𝑧∈𝑈), (30) which implies that 1)ℎ(𝑧)1−𝜌(1−𝜁+𝜁²+)ℎ(𝑧)×()(1−𝜁+𝜁2𝛾1−𝜁𝑧ℎ^′(𝑧)𝛿().𝛼+𝛽−𝜌∈𝑃𝑖=1,2 (31) We form the functional Φ(𝑢,𝑣) by choosing 𝑢=ℎ_𝑖(𝑧), 𝑣=𝑧ℎ^′_𝑖(𝑧), that is, Φ(𝑢,𝑣)=(1−𝜁)𝑢+𝜁²+)()𝑣(1−𝜁𝑢+𝜁2𝛾1−𝜁𝛿𝛼+𝛽−𝜌. (32) Clearly, the conditions (i) and (ii) of Lemma 3 are satisfied. Now, we verify the condition (iii) as follows: ℜΦ𝑖𝑢₂,𝑣₁=𝜁²−(1−𝜁)²𝑢²₂+2𝛾(1−𝜁)𝑣₁𝛿𝛼+𝛽−𝜌≤𝜁²−𝜌−(1−𝜁)²𝑢²₂−𝛾𝜁(1−𝜁)1+𝑢²₂𝛿=𝛼+𝛽𝐴+𝐵𝑢²₂,2𝐶 (33) where 𝐴=𝜁²𝛿(),)𝛼+𝛽−𝛾𝜁1−𝜁−𝜌𝛿𝛼+𝛽𝐵=−(1−𝜁²(,𝛿+𝛾𝜁1−𝜁)𝐶=𝛼+𝛽2. (34) We note that ℜ{Φ(𝑖𝑢₂,𝑣₁)}<0 if and only if 𝐴=0, 𝐵<0. From 𝜁 as given by (28), we have 0≤𝜁<1, 𝐴=0, and 𝐵<0. Therefore, applying Lemma 3, we have ℎ_𝑖(𝑧)∈𝑃(𝑖=1,2) and consequently 𝑀(𝑧)∈𝑃_𝑘(𝜁)for 𝑧∈𝑈. This completes the proof of Theorem 6.

Similarly, we can prove the following theorem for the class 𝑀_𝑝(𝛼,𝛽,𝜇,𝛿,𝜌,𝛾).

Theorem 7. If 𝑓(𝑧)∈𝑀_𝑝(𝛼,𝛽,𝜇,𝛿,𝜌,𝛾), then 𝑧^𝑝𝐻^𝛼_{𝑝,𝛽,𝜇}𝑓(𝑧)^𝛿/2∈𝒫_𝑘(𝜆), (35) where 𝜆 is given by 𝜆=𝛾+𝛾²+𝜌𝛿𝜇2𝛾+𝛿𝜇.2𝛾+𝛿𝜇 (36)

Remark 8. Putting 𝜇=1, in Theorems 4 and 6, we obtain the corresponding results for the operator 𝐻^𝛼_𝑝,𝛽 defined in (14).

Acknowledgment

The authors would like to thank the referees of the paper for their helpful suggestions.